Turbo code interleaver with near optimal performance

ABSTRACT

A method of interleaving blocks of indexed data of varying lengths is disclosed. The method includes the steps of: providing a set of basic Interleavers comprising a family of one or more permutations of the indexed data and having a variable length; selecting one of the basic Interleavers based upon a desired Interleaver length L; and adapting the selected basic Interleaver to produce an Interleaver having the desired Interleaver length L.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.11/980,917, filed Oct. 31, 2007, currently pending, which is acontinuation of U.S. application Ser. No. 11/051,585, filed Mar. 30,2005, now U.S. Pat. No. 7,526,687, issued Apr. 28, 2009, which is acontinuation of U.S. application Ser. No. 10/024,834, filed Dec. 19,2001, now U.S. Pat. No. 6,925,587, issued Aug. 2, 2005, which is adivisional of U.S. application Ser. No. 09/375,067, filed Aug. 16, 1999,now U.S. Pat. No. 6,334,197, issued Dec. 25, 2001, which claims thebenefit of U.S. Provisional Application Ser. No. 60/096,807, filed Aug.17, 1998.

BACKGROUND OF THE INVENTION

The present invention relates to error correction in coding schemes fordigital communication systems, and more particularly to designoptimization for Interleavers of any size within a specified wide rangeused in error correction. Even more particularly, the present inventionrelates to optimization of Turbo Interleavers such that smaller optimalInterleavers can be built from larger optimal Interleavers.

Interleaving is a process of reordering a sequence of symbols or bits ina predetermined manner. “Interleaver size” is equal to the size of thesequence. The apparatus performing the interleaving is referred toherein as an Interleaver.

Turbo Interleavers are interleavers used in the construction of turbocodes. In a turbo code built as a parallel concatenation of twoconstituent recursive convolutional codes, a Turbo Interleaver serves toreorder an input data sequence in a pseudo-random fashion prior to anencoding by a second of the constituent codes. As a result, separateencodings produced by the two constituent encoders are largelyuncorrelated, which property allows them to be combined by a turboencoder to produce a composite encoding with excellent error protectioncapability.

S-random Interleavers are one of the most widespread forms of turboInterleavers.

The principle behind S-random Interleavers is to avoid mapping neighborpositions of an original input sequence to another neighbor position ofthe interleaved sequence within a window of size s. The design goal inS-random Interleavers is to maximize S while preserving the aboveprinciple. However, S-random Interleavers have to be re-designed everytime the Interleaver size is changed and there is typically norequirement of any resemblance between the Interleavers with similarsizes.

Thus, it is desirable to have a general

Interleaver design for Interleavers of any size within a set of sizes,wherein the design methodology is concise and efficient such that thesame Interleaver design is near-optimal for all Interleavers within theset of sizes. It is also advantageous to have a design for building anear-optimal Interleaver that can easily be reduced to smaller-sizednear-optimal Interleavers without performance degradation.

Therefore, the present invention advantageously addresses the above andother needs.

SUMMARY OF THE INVENTION

The present invention advantageously addresses the needs above as wellas other needs by providing a method and apparatus for a turboInterleaver which employs Interleavers of variable length employing oneor more permutations.

In one embodiment, the invention is characterized as a method ofinterleaving blocks of indexed data of varying length. The methodincludes the steps of: providing a set of basic Interleavers comprisinga family of one or more permutations of the indexed data and having avariable length; selecting one of the basic Interleavers based upon adesired Interleaver length L; and adapting the selected basicInterleaver to produce an Interleaver having the desired Interleaverlength L.

In another variation, a method of interleaving blocks of indexed data ofvariable length includes the steps of: providing a family of basicInterleavers comprising “two-dimensional permutations” includingcomputing the “two-dimensional permutations”, further comprising:writing the indexed data into an Interleaver matrix having one or morerows in each of two dimensions; permuting the indexed data in one ormore rows in at least one of the two dimensions to produce “constituentpermutations”, possibly being different from one row to another row,wherein the constituent permutations are pseudo-random permutationsdescribed by a limited number of parameters, wherein an amount ofstorage required for storing the limited number of parameters is lessthan that for storing a vector representation of the constituentpermutations; reading out the data from the Interleaver matrix;selecting one of the basic Interleavers for use in encoding based upon adesired Interleaver length L; adapting the selected basic Interleaver toproduce an Interleaver having the desired Interleaver length L; whereinthe selecting includes: identifying a group of the basic Interleavershaving a length greater than or equal to the desired Interleaver lengthL; and selecting one of the basic Interleavers having a length smallestamong the identified group of the basic Interleavers; wherein theadapting includes: deleting indexed data having indices higher thanrequired for a permutation of length L; providing an Interleaver devicefor interleaving blocks of indexed data, the Interleaver device furthercomprising a memory device for storing descriptions of the basicInterleavers; and storing the descriptions in the memory device.

In another embodiment, a system for interleaving and turbo encodingblocks of indexed data of varying length, comprises: a parallelconcatenation of two or more constituent encoders for recursiveconvolutional codes of recursion period p; and an Interleaver devicecoupled to the parallel concatenation for performing the steps of:accessing stored descriptions of basic Interleavers, the basicInterleavers comprising a family of one or more permutations of theindexed data and having a variable length; identifying a group of thebasic Interleavers having a length greater than or equal to a desiredInterleaver length L; selecting one of the basic Interleavers having alength which is smallest among the group of the basic interleaves; andadapting the selected one of the basic Interleavers to produce anInterleaver having the desired Interleaver length L.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features and advantages of the presentinvention will be more apparent from the following more particulardescription thereof, presented in conjunction with the followingdrawings wherein:

FIG. 1 is a block diagram of hardware of a mobile communication systemof a type that could be used to implement the teachings of the presentinvention;

FIG. 2 is a functional block diagram of a Turbo encoder which could beimplemented in the system of FIG. 1;

FIG. 3 is a flow chart of steps traversed by the mobile communicationsystem of FIG. 1 and encoding system of FIG. 2 in accordance with anembodiment of the present invention;

FIG. 4 is a diagram of performance curves of Turbo Interleavers such asshown in FIG. 2 of Size 1024 bits at Code Rate ½, using four (4) decoderiterations comparing Galois Field Interleavers to S-Random Interleaversand to Random Interleavers for Bit Error rate (BER) and Frame Error Rate(FER) performances;

FIG. 5 is a diagram of performance curves of a Turbo Interleaver such asshown in FIG. 2 of Size 1024 at Code Rate ½, using eight (8) decoderiterations comparing Galois Field Interleavers to S-Random Interleaversand to Random Interleavers for Bit Error Rate (BER) and for Frame ErrorRate (FER); and

FIG. 6 is a diagram of performance curves of a Turbo Interleaver such asshown in FIG. 2 of size 1152 bits at Code Rate ⅓, using four (4) decoderiterations comparing Galois Field Interleavers to S-Random Interleavers,Random Interleavers for Bit Error Rate (BER) and for Frame Error Rate(FER).

Corresponding reference characters indicate corresponding componentsthroughout the several views of the drawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The following description of the presently contemplated best mode ofpracticing the invention is not to be taken in a limiting sense, but ismade merely for the purpose of describing the general principles of theinvention. The scope of the invention should be determined withreference to the claims.

Referring to FIG. 1, a block diagram is shown of a digital communicationsystem using Turbo Codes of a type that could be used to implement theteachings of the present invention. It comprises transmitter hardwareincluding: a Transmitter Interface 102, an A/D Converter 104; aSegmentation Processor 106; a Turbo Coder 108; a Burst Formatter 110; aModulator 112; a Transmitter (RF/IF) 114. It also comprises a PowerSupply 124; a Timing and Control Processor 116; a Synthesizer andOscillator 118; and a Switch 120. It comprises receiver hardwareincluding: a Receiver Interface 134; a Turbo Decoder 132; an Equalizer130; a Receiver/Demodulator 128; and a Receiver (RF/IF) Preamp Mixer126.

The transmitter receives an analog signal through a TransmitterInterface 102 and performs an A/D conversion at A/D Connector 104. Thediscrete samples generated from the A/D Converter 104 are fed toSegmentation Processor 106 where fixed-length data units of 44 octetsare formed by fragmenting an initial MAC protocol data unit (IMPDU), andthen the fixed length data units passed to the Turbo Coder 108 whichuses an

Interleaver to pseudo-randomize the input between 2 concatenatedencoders and encodes the fixed length data units (data streams) andsends encoded data units (data streams) to the Burst Formatter 110.

A burst, a series of repetitive waveforms at a prescribed time andamplitude lasting a short duration, is formed at Burst Formatter 110 andis passed to Modulator 112 where the burst is modulated by mixing with acarrier waveform of known frequency. Transmitter 114 transmits themodulated burst when the switch 120 connects antenna 122 to theTransmitter 114. The Synthesizer and Oscillator 118 keeps track oftiming for the transmitter (RF/IF) 114 and for the Timing and ControlProcessor 116 which controls when bursts are formatted.

When the Antenna 122 receives a burst and the receiver (RF/IF) PreampMixer 126 is connected to the antenna through the switch 120, thereceived burst is amplified by the receiver (RF/IF) Preamp Mixer 126,and then demodulated to remove the carrier waveform frequency. TheEqualizer 130 filters the demodulated burst with filters adjusted so asto produce an enhanced digital signal which is next Turbo decoded byTurbo Decoder 132 through a concatenation of decoders and an Interleaverusing feedback from each other decoder to decode information from thereceived burst. Decoded data is converted from digital to analog by D/AConverter 134 and passed through receiver interface 136 to anothersystem for further processing as needed. Since the digital communicationsystem of FIG. 1 would typically communicate using a variety ofdifferent information block sizes depending on the service requirementssuch as for voice or packet data, the embedded turbo code interleavermust be flexible enough to accommodate multiple block sizes withoutundue sacrifice in turbo code performance.

Referring to FIG. 2, a functional block diagram is shown of arepresentative turbo code encoder consisting of a parallel,concatenation of two simple constituent encoders (encoders) 10, 10′,coupled to an Interleaver with memory (Interleaver) 16 and a puncturer36. The first encoder 10 comprises: modular adders (or binary adders)17, 20, 26, 28, 24, 25, and 30; shift register delay elements (or “shiftregisters”) 18, 21, 22; a switch 12; output connections for aninformation bit X(t) and for parity bits Y₀(t), Y₁(t). The secondencoder 10′ comprises analogous hardware 17′, 20′, 26′, 28′, 24′, 25′,30′, 18′, 21′, 22′, 12′. Output X(t) is coupled to the switch 12 coupledto input X(t). Output Y₀(t) is coupled to modular adder 24 coupled tomodular adder 20 at its output, which is coupled to register 18 andmodular adder 17 at its input, which is coupled to switch 12. OutputY₁(t) is coupled to modular adder 25 coupled to modular adder 28 at itsoutput and to register 22 at its output. Modular adder 28 is coupled tomodular adder 26 at its output and register 21 at its output; modularadder 26 is coupled to modular adder 17 at its output and to register 18at its output. Modular adder 30 is coupled to modular adder 17 at itsinput. A detailed description of how the Turbo Coder of FIG. 2 operatesin practice follows.

The two constituent encoders 10, 10′ produce parity bits Y₀(t), Y₁(t)and Y₀′(t), respectively, selected ones of which are removed from anoutput stream (output) of the two simple constituent encoders 10, 10′according to a prescribed puncturing pattern by the puncturer 36 inorder to achieve a desired overall Turbo code rate. Both the firstencoder (encoder #1) 10 and the second encoder (encoder #2) 10′ processthe same information bit stream and X(t) (or “information bits”), butthe encoder #2 10′ processes information bits X(t) in a different orderthan the order in which encoder #1 10 does since the Interleaver 16processes the information bits X(t) before they reach encoder #2 10′. Byrearranging an order of presentation of the information bits X(t), theInterleaver 16 serves to decorrelate the outputs of the two simpleconstituent encoders 10, 10′ so that the information bits X(t) causingencoder #1 10 to produce a low-Hamming weight output are unlikely tocause encoder #2 10′ to also produce a low-Hamming weight output.

In FIG. 2, the Interleaver 16 avoids mapping a “neighbor position” to acorresponding “neighbor position” of the interleaved bit sequence. TheInterleaver 16 does this in a pseudo-random fashion by re-ordering bitlocations in a random-looking predetermined fashion.

Both encoders 10, 10′ produce, in addition to the information bitsX(t)(or systematic bits), parity bits Y₀(t) and Y₁(t) which arepunctured by puncturer 36 to achieve a desired overall Turbo Code rate.

Information bit stream X(t) is received at switch 12, and is processedin accordance with several modular adders of above and shift registersabove which are hard-wired to represent two (2) numerator polynomialsand one denominator polynomial.

Referring still to FIG. 2, a denominator polynomial d(D), representingTurbo Code “1010”, is hardwired by the return feedback connection tomodular adder 17 and its respective connections to modular adder 30.Before computing, three shift registers 18, 21, and 22 are first zeroed.

A first numerator polynomial over a denominator polynomial, representingTurbo Code “1101” is hardwired to return output Y₀(t) by combining: X(t)with a result of modulator adder 17 to create a first bit W(t); themodular sum (second bit) of shift register 18 and W(t) from the modularadder 20; another zero bit (third bit) indicated by the lack ofconnection to the register 21;

and the modular sum (fourth bit) of another register 22 and a result ofmodular adder 20 from modular adder 24. The result isY₀(t)=W(t)+S₀(t)+S₂(t).

Information bit stream X(t) is presented in its original, uninterleavedorder at a switch 12 and processed by the first encoder 10. In FIG. 2,the first encoder 10 is implemented as a linear feedback shift registerwhose transfer function is:

${G(D)} = {\left\lbrack {1\frac{1 + D + D^{3}}{1 + D^{2} + D^{3}}\frac{1 + D + D^{2} + D^{3}}{1 + D^{2} + D^{3}}} \right\rbrack.}$

Thus, during an encoding step at time t≧0, a shift register contents ofthe shift register 18, 21, 22, are S₀(t), S₁(t), S₂(t) and theinformation bit X(t) is present at the input to binary adder 17. Theencoder 10 then produces its two coded output bits (coded bits) Y₀(t),Y₁(t) according to the following two summations:

Y ₀(t)=W(t)+S ₀(t)+S ₂(t)

Y ₁(t)=W(t)+S ₀(t)+S ₁(t)+S ₂(t),

wherein

W(t)=X(t)+S ₁(t)+S ₂(t).

After the coded bits are output, the current encoding step at time t iscompleted by shifting the contents of the shift register 18, 21, 22 onceto prepare for a next encoding step at time t+1. At time t+1: S₀(t+1)=W(t), S₁(t+1)=S₀(t) and S₂(t+1)=S₁(t). At the start of anencoding process at t=0, the shift register contents are initialized tozero, wherein S₀(0)=S₁(0)=S₂(0)=0. The second encoder 10′ operates inthe same fashion on an output of Interleaver 16 to produce another two(2) coded output bits.

Since the digital communication system of FIG. 1 would typicallycommunicate using a variety of different information block sizesdepending on the service requirements such as for voice or packet data,an embedded turbo code interleaver within turbo coder 108 must beflexible enough to accommodate multiple block sizes without unduesacrifice in turbo code performance. In its most general form, aninterleaver design proposed herein consists of a collection of basicinterleavers of various block lengths, an algorithm for selecting one ofthe basic interleavers to use as a “mother” interleaver, and a method ofadapting the mother interleaver to produce a turbo interleaver of aparticular desired length.

The basic interleavers are stored in a memory, which may be locatedwithin the Interleaver 16 as in FIG. 2, either as an explicit table ofread or write indices or as a smaller set of parameters from which thetable of read or write indices can be regenerated according to apredetermined algorithm.

A couple of simple examples will clarify these concepts. First, consideran interleaver of length 8 using a permutation π=(04261537). Thispermutation could be used as either a list of write (input sequence)addresses or read addresses. Let d₁(0), d₁(1), d₁(7) denote input data(input sequence) in their original sequence; and let d_(o)(0), do(1), .. . , d_(o)(7) denote values of the same input data but in a permutedorder. The interleaver could be implemented to write d₁(0) to outputposition 0, d₁(1) to output position 4, d₁(2) to output position 2,d₁(3) to output position 6, etc. In this case, the interleaver actioncan be expressed mathematically as

d _(o)(π(k))=d ₁(k)

Alternately, the interleaver could be implemented to read data valuesfrom the input data according to the permutation π. That is, a firstinterleaved value d_(o)(0) is read from input position 0, a secondinterleaved value d_(o)(1) is read from input position 4, and so on.Mathematically,

d _(o)(k)=d ₁(π(k)).

Neither interpretation is to be preferred; it is merely a matter ofconvention. For purposes of describing interleaver operations herein,the first interpretation (in which permutations specify write addressesfor the interleaver) is used.

It should be noted, however, that in a turbo decoder, such as the turbodecoder 132 in FIG. 1, both interleaving and its inverse(de-interleaving) are used. If the interleaver is implemented to use thepermutation π as write addresses, the de-interleaver can be implementedto use the permutation π as read addresses. This means that theinterleaving and de-interleaving operations can share the samepermutation generation hardware of software. It is not necessary tostore descriptions of both the interleaver and its de-interleaverseparately.

The permutation π=(04261537) arises from bit-reversal indexing. Forexample, input position 1 has a 3-bit binary representation 001 and ismapped to output position 4, which has 100 as its 3-bit binaryrepresentation. Likewise, input position 3 (binary 011) is mapped tooutput position 6 (binary 110). In VLSI hardware or on some digitalsignal processing, which, optionally may be employed within theInterleaver 16 bit-reversed indexing is easily accomplished withoutspecial memory storage.

The-permutation π=(03614725) can be generated by the simple mathematicalrecursion:

α=3; π(0)=); π(k)=π(k−1)+α(mod 8).

Different recursions of this type can be described by the two parametersα and π(0). Thus, a family of basic interleavers based on simplerecursive formulas could be represented by a small table of parametersstored in the memory. Advantageously, for an interleaver of large blocklength or a large set of interleavers of various block lengths, theability to store a small table of parameters rather than the explicitpermutations results in a large reduction in memory requirements. Thus,it is advantageous to design turbo interleavers in this way provided aparameterized family of interleavers results in good turbo codeperformance. These design issues are favorably addressed by the proposedinvention.

Given a family of basic interleavers of various block sizes representedand stored in memory in some fashion, a turbo device (either the turboencoder 108 or the turbo decoder selects one of them for use inimplementing an interleaver of specific length L. In one preferredembodiment of the invention, the lengths of the basic interleavers areall different, and the turbo device selects the basic interleaver havinga smallest length N among all basic interleavers whose lengths are atleast as big as the desired length L.

In other embodiments, it may be desirable to have multiple basicinterleavers all of the same length. For example, there may be animplementation advantage in having all basic interleavers have lengthsthat are integral powers of two. In such a design, there may be multiplebasic interleavers of length N=2^(c), each optimized for a differentinterval of block sizes between 2^(c−1) and 2 ^(c). In such embodiments,the turbo device (turbo encoder 108, turbo decoder 132) first identifiesa set of basic interleavers having a smallest length N among all thebasic interleavers whose lengths are at least as big as the desiredlength L and then selects one of the basic interleavers in the setaccording to other selection criteria depending on L.

Once a basic interleaver has been selected, it is then adapted to lengthL by the process referred to herein as pruning. Pruning refers to adiscarding of permutation indices that are invalid for a pruned matrix.For example, one prunes the permutation π=(03614725) f length 8 on theintegers modulo 8 to a new permutation of length 5 on the integersmodulo 5 by ignoring the invalid indices 5, 6 and 7. Thus, a prunedpermutation is π*=(03142).

The process of pruning, in accordance herewith, is further explained byan algorithm shown in FIG. 3. For simplicity, the algorithm assumes thatthe basic interleavers all have lengths that are integral multiples oftwo. The processing steps are as follows:

The above rules are refined later herein so that Rule 1 and Rule 2continues to be satisfied for Turbo Interleavers of any size N obtainedfrom a single Interleaver of size 2^(m) by means of puncturing(2^(m−1)<N≦2^(m)). Obtaining an Interleaver of any size N from a motherInterleaver of a larger size via puncturing is one aspect of thisinvention.

A smaller Interleaver I_(N) _(s) of size N_(s) is formed by using apre-designed Interleaver matrix, I₂ _(m) , of size 2^(m), where m ischosen such that it is the smallest integer for which 2^(m)≧N_(s), i.e.,the smallest power of two that is larger than or equal to the size Nnumber of elements, N an integer, in the Interleaver I_(N).

A Smaller Interleaver, I_(N) _(s) , is then generated from thepre-designed Interleaver, I₂ _(m) , by puncturing the predesignedInterleaver, I₂ _(m) .

Thus, a Smaller Interleaver, I_(N) _(s) , is created by only acceptingbit positions into the smaller Interleaver, I_(N) _(s) , from theoriginal pre-designed Interleaver, I_(s) _(m) , if the bit positionvalue is smaller than the size of the smaller Interleaver, I_(N) _(s) ,measured by the number of elements in the smaller Interleaver, N_(s).

This can be accomplished, for example, by a processor modified with acomputer program, the steps of which are shown in FIG. 3, that initiatesthe following steps:

1) Initialize a counter i to zero, where i represents a new smallerInterleaver bit position, and j represents an original largerInterleaver bit position. This corresponds to Initialize Counter 310 inFIG. 3;

2) For every original bit position I₂ _(m) [j]; where j is from 0 to2^(m)−1 (Check j 320 of FIG. 3), initiate the further steps:

If I₂ _(m) [j]<N_(s), set I_(N) _(s) [i]=I₂ _(m) [j] and increment thecounter i. These steps correspond to Check Larger I Element 330, SetSmaller I Element 340, and Increment Counter 350 respectively, of FIG.3.

Otherwise reject I₂ _(m) [j] per Reject and Return 360 of FIG. 3. Thisprogram accepts, consecutively, from a first to a last bit position, anyoriginal bit position of an original Interleaver which has a value lessthan the smaller Interleaver size.

Pruning is a key aspect of the invention described herein. It is theadvantage that the method is easily implemented in either a VLSI or aDSP and so provides an efficient mechanism for providing interleavers ofarbitrary lengths without storing separate descriptions for everypossible length. The set of basic interleavers are designed to be robustwith respect to pruning in accordance with principles to be described inconjunction with a detailed, explicit design illustrative of theinvention.

The design of a turbo interleaver should take into account the structureof the constituent recursive convolutional codes in order to ensure thatan overall Turbo code has a favorable Hamming weight spectrum (“weightspectrum”) leading to good error correction performance. The weightspectrum of a linear binary code of length N is a tabulation giving anumber of code words of each Hamming weight from 0 to N. A Hammingweight is a number of non-zero entries in the code word. Since theconstituent code of a Turbo Code is recursive, it takes an inputsequence of a Hamming weight of at least two (2) to cause a systematicencoder to leave an all zero state and later to return to the all zerostate and therefore to generate a less desirable, low parity Hammingweight sequence. In general, a systematic, recursive encoder wouldgenerate a parity sequence of high Hamming weight for input sequenceshaving a Hamming weight of one (1), since the encoded sequence uponleaving the all zero-state can never return to it. For recursiveconvolutional codes as constituent codes, the probability that bothencoders generate encoded sequences that leave the all zero-state andlater return to the all zero-state is the highest when the inputsequence is of Hamming weight two (2).

It is also observed that when the recursive eight-state constituentencoders have a primitive feedback polynomial of degree 3, an inputsequence of Hamming weight two (2) can cause the first constituentencoder to generate a finite error event only if the two “1's” in theinput sequence are separated by 6+7n (n is an integer) zeros. It istherefore important that any input sequence consisting of exactly two1's separated by 6+7n (n∈N) zeros should not be mapped by theinterleaves to a new sequence with two 1's now separated by 6+7m (m∈N)zeros. In that way the second encoder 10′ will generate high parityHamming weight when the first encoder generates low parity Hammingweight, and vice versa, corresponding to an input sequence of Hammingweight two (2).

Even if the two 1's are separated by the undesirable 6+7n zeros in aninput sequence of Hamming weight two, the corresponding parity Hammingweight will grow larger as n grows larger. Thus, it is less crucial toaddress the cases for n>l, as the most critical values for n are 0followed by 1, respectively. This is because as n grows, parity Hammingweight grows sufficiently larger.

The rules that are introduced in one embodiment of the invention todesign Turbo Interleavers for eight-state Turbo codes are thus:

Rule 1: Minimize the occurrence of events:

|I[x]−I[x−7]|=7   (1)

wherein I[x] denotes the position that x is mapped to by the Interleavermatrix I.

Rule 2: If the first rule is satisfied with zero occurrences of equation(1), minimize the occurrence of event:

|I[x]−I[x−7]|=14, or   (2)

|I[x]−I[x−14]|=7, or

|I[x]−I[x−14]|=14

By following the above created rules, the probability of both of theencoders 10, 10′ generating low-Hamming weight parity sequences isminimized.

An explicit exemplary turbo interleaver design (exemplary design) willnot be described in order to more fully illustrate and develop theconcepts of the invention. In the exemplary design, each of the basicinterleavers implemented by the Interleaver 16 is a two-dimensionalblock interleaver (or interleaver matrix) of dimension R×C, whereR−2^(c) is a number rows and C−2^(c) is a number of columns.Conceptually, the input data (data) are written into the interleavermatrix row by row. Then row and column permutations are performed torandomize data positions. The data are then read out column by column.Specifically, given an input position l=c·i+j, a corresponding outputinterleaved position will be given by mathematical formulaI(l)=R·π_(i)(j)+ρ(i), wherein π_(i) is a column permutation applied todata in row i and wherein ρ is bit-reversed indexing, which isespecially simple to implement and requires no additional parameterstorage in the memory of the interleaver 16.

One could, of course, make the p permutation different for differentcolumns ad the expense of additional implementation complexity andincreased storage requirements to specify each of the individual columnpermutations. In either case, the ρ premutation(s) should performpseudo-random interlacing of top and bottom halves of the interleavermatrix in order to facilitate on-the-fly implementation of pruning. Suchinterlacing ensures that, if I(l) is an invalid index for a prunedinterleaver, then I(l+1) will be a valid index, assuming that no basicinterleaver is pruned to half its length or beyond.

The proposed two-dimensional structure is advantageous for use in turbointerleaving for several reasons. Since the turbo interleaver is builtin a structured way from simple constituent permutations that can bedescribed by a small set of parameters, implementation complexity issmall. When different constituent permutations are used from row to row,a composite interleaver permutation exhibits sufficient randomness toachieve good turbo code performance despite its low complexity.Furthermore, by choosing R and C appropriately, one can balance the“spreading capability” of the interleaver (how well is separatesneighboring positions) and its “randomness” properties. The spreadingcapability of the interleaver is also important for the turbointerleaver in that it helps to enhance the overall “weight spectrum” ofthe turbo code. In general, spreading capability increases withincreasing R, and randomness increases with increasing C.

Preferably, as a rule of thumb, for building interleaver matrixes inaccordance with the invention, one would make R as large as possiblewithout making C so small that the randomness produced by thepermutations applied to each row is degraded.

Thus, in the illustrative designs presented below, the set of basicinterleavers have the property that, in general, those of larger lengthuse a larger number of rows R. This is an important aspect of thetwo-dimensional design.

In the illustrative designs, in accordance with one aspect of theinvention presented below, the constituent permutations applied to rowsof the interleaver matrix are based on a novel class of permutationsderived from Galois field arithmetic.

A Galois Field (GF) with p^(m) elements is denoted as GF(p^(m)), whereinp is a prime number and m is any integer greater than one (1). It can beformed from GF(p) using a primitive polynomial p(x) of degree m overGF(p)[x]. In the case of GF(2^(m)), the roots of primitive polynomialp(x) of degree m over GF(2)[x], form a subset of the primitive elementsin GF(2^(m)). A primitive element in a Galios field with q elements hasorder q−1, i.e., the smallest positive integer n such that α^(m)=1 isn=q−1.

If ∝ is a primitive element in GF(2^(m)), all of the other nonzeroelements of GF(2^(m)) can be obtained as consecutive powers of α.

GF(2^(m))={0, α⁰=1, α, α², α³, . . . , α^(2m−2)}  (5)

Furthermore, every element of the field GF(2^(m)) can be expressed interms of 1, α, α², . . . , α^(m−1);

For example, GF(8) can be constructed from GF(2) using the primitivepolynomial p(x)=x³+x+1 over GF(2)(x). Let a be a root of p(n).Multiplication of elements can be performed using the fact that α⁷=1 (bydefinition since α is primitive in GF(8)). Addition of elements in GF(8)can be performed using equalities in terms of 1, α, and α² since α³=α+1in Galois Field arithmetic (where 1≡−1).

An exemplary multiplication of elements in GF(8) is Equation (4).

α³·α⁶=α⁹=α²   (6)

An exemplary addition is Equation (5).

α³+α⁴=(α+1)+(α²+α)=α²+1=α⁶   (7)

An Interleaver of size 2^(m), I₂ ^(m), is formed by the following four(4) steps:

(1) First, a matrix is filled row by row with bit positions startingwith 0 in the upper leftmost position, and ending with (rxc−1, where2^(m)=rxc as defined above) in the lower rightmost position. This is theconventional manner of filling Interleaver matrices.

Thus, an Interleaver matrix of size 32=4×8 would result in matrix (8):

$\begin{matrix}\begin{bmatrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\24 & 25 & 26 & 27 & 28 & 29 & 30 & 31\end{bmatrix} & (8)\end{matrix}$

(2) Secondly, permute each row i (i=0, 1, 2, . . . , r−1) (within itselfaccording to a predetermined rule. One method is to permute) accordingto the following permutation rules employing Galois Field arithmetic(9):

j→log_(α) i _(b)(α^(i) ^(o) +α^(j)) for j=0, 1, 2, 3, . . . , c−2

j→log_(α) i _(b)(α^(i) ^(o) ) for j=c−1   (9)

In Equation (9), ∝ is a root of the primitive polynomial p(n) used toconstruct GF(c), α^(i) ^(b) is primitive in GF(c) and i₀ is a designedinteger between 0 and c−2 inclusive, and i_(b) is a predeterminedinteger, i₀ and i_(b) selected based upon certain design rules to bedescribed.

Furthermore, by definition log_(α)i_(b)(o) is set to (c−1) as a resultof the second part of Equation (9).

An exemplary permutation would be such as is shown in Equation (10) foreach row i (i=0, 1, 2, 3), for an Interleaver of size 32 having 8columns and 4 rows.

j→log_(α) i _(b)(α^(i) ^(o) +α^(j)) for j=0, 1, 2, 3, 4, 5, 6

j→log_(α) i _(b)(α^(i) ^(o) ) for j=7   (10)

For the sake of demonstration, Table 1 is constructed with constantsi_(b and) i₀ to permute a Turbo Interleaver matrix of size N_(s), withinthe set N, where 16<N≦32. The values of Table 1 are fabricated hereinfor the sake of the following example for the construction of I³².

TABLE 1 Constants i i_(b) i₀ 0 1 0 1 1 2 2 3 5 3 6 4

Thus, in accordance with Table 1, each row is shuffled such that:

$\begin{matrix}{{{{{row}\mspace{14mu} i} = {\left. {0\text{:}\mspace{11mu} j}\leftarrow{{\log_{\alpha}\left( {\alpha^{j} + 1} \right)}\mspace{14mu} {for}\mspace{14mu} j} \right. = 0}},1,2,3,4,5,6}{\left. j\leftarrow{{\log_{\alpha}(1)}\mspace{14mu} {for}\mspace{14mu} j} \right. = 7}} & (11) \\{{{{row}\mspace{14mu} i} = {\left. {1\text{:}\mspace{11mu} j}\leftarrow{{\log_{\alpha}\left( {\alpha^{j} + \alpha^{2}} \right)}\mspace{14mu} {for}\mspace{14mu} j} \right. = 0}},1,2,3,4,5,{\left. {6j}\leftarrow{{\log_{\alpha}\left( \alpha^{2} \right)}\mspace{14mu} {for}\mspace{14mu} j} \right. = 7}} & (12) \\{{{{{row}\mspace{14mu} i} = {\left. {2\text{:}\mspace{11mu} j}\leftarrow{{\log_{\alpha}^{3}\left( {\alpha^{j} + \alpha^{5}} \right)}\mspace{14mu} {for}\mspace{14mu} j} \right. = 0}},1,2,3,4,5,6}{\left. j\leftarrow{{\log_{\alpha}^{3}\left( \alpha^{2} \right)}\mspace{14mu} {for}\mspace{14mu} j} \right. = 7}} & (13) \\{{{{{row}\mspace{14mu} i} = {\left. {3\text{:}\mspace{11mu} j}\leftarrow{{\log_{\alpha}^{6}\left( {\alpha^{j} + \alpha^{4}} \right)}\mspace{14mu} {for}\mspace{14mu} j} \right. = 0}},1,2,3,4,5,6}{\left. j\leftarrow{{\log_{\alpha}^{6}\left( \alpha^{4} \right)}\mspace{14mu} {for}\mspace{14mu} j} \right. = 7}} & (14)\end{matrix}$

The shuffling of each row results in a pseudo-random order of positionsrepresented by sequences to the right of each of the arrows in equations(15), which represent the original bit positions which map to newlyordered sequences to the left of each arrow:

$\begin{matrix}\begin{matrix}{{{row}\mspace{14mu} 0\text{:}\mspace{14mu} \left( {0,1,2,3,4,5,6,7} \right)} - \left( {7,3,6,1,5,4,2,0} \right)} \\{{{row}\mspace{14mu} 1\text{:}\mspace{14mu} \left( {0,1,2,3,4,5,6,7} \right)} - \left( {6,4,7,5,1,3,0,2} \right)} \\{{{row}\mspace{14mu} 2\text{:}\mspace{14mu} \left( {0,1,2,3,4,5,6,7} \right)} - \left( {6,2,1,3,0,7,5,4} \right)} \\{{{row}\mspace{14mu} 3\text{:}\mspace{14mu} \left( {0,1,2,3,4,5,6,7} \right)} - \left( {2,5,6,1,7,0,4,3} \right)}\end{matrix} & (15)\end{matrix}$

The above shuffling results in an Interleaver matrix:

$\begin{matrix}\begin{bmatrix}7 & 3 & 6 & 1 & 5 & 4 & 2 & 0 \\14 & 12 & 15 & 13 & 9 & 11 & 8 & 10 \\22 & 18 & 17 & 19 & 16 & 23 & 21 & 20 \\26 & 29 & 30 & 25 & 31 & 24 & 28 & 27\end{bmatrix} & (16)\end{matrix}$

(3) Thirdly, each of the rows of Interleaver matrix I₃₂ are shuffled orre-ordered according to any method that interlaces an upper half of thematrix resulting from the above permutations, with a lower half of thematrix.

One method of doing this is to re-order the rows according to a bitreversal on row index (e.g. as represented by the pattern (00,01,10,11)for a four-row matrix). From the above example matrix, this results inmatrix:

$\begin{matrix}\begin{bmatrix}7 & 3 & 6 & 1 & 5 & 4 & 2 & 0 \\22 & 18 & 17 & 19 & 16 & 23 & 21 & 20 \\14 & 12 & 15 & 13 & 9 & 11 & 8 & 10 \\26 & 29 & 30 & 25 & 31 & 24 & 28 & 27\end{bmatrix} & (17)\end{matrix}$

(4) Fourthly, the contents of the resulting permuted and re-orderedmatrix, or Interleaver matrix, are read out column by column to anencoder as in the case of a Block Interleaver.

In the above example, this results in the bit position sequence:

$\begin{matrix}{7\mspace{14mu} 22\mspace{14mu} 14\mspace{14mu} 26\mspace{14mu} 3\mspace{14mu} 18\mspace{14mu} 12\mspace{14mu} 29\mspace{14mu} 6\mspace{14mu} 17\mspace{14mu} 15\mspace{14mu} 30\mspace{14mu} 1\mspace{14mu} 19\mspace{14mu} 13\mspace{14mu} 25\mspace{14mu} 5\mspace{14mu} 16\mspace{14mu} 9\mspace{14mu} 31\mspace{14mu} 4\mspace{14mu} 23\mspace{14mu} 11\mspace{14mu} 24\mspace{14mu} 2\mspace{14mu} 21\mspace{14mu} 8\mspace{14mu} 28\mspace{14mu} 0\mspace{14mu} 20\mspace{14mu} 10\mspace{14mu} 27} & (18)\end{matrix}$

Permutations of the rows within themselves should be done in such a waythat Rule 1 and Rule 2 are satisfied for any Interleaver size N obtainedfrom an original Interleaver of size 2^(m) where 2^(m−1)<N≦2^(m).

The preferred basic interleaver structure, described previously hereinallows the formulation of simple design criteria that help ensurerobustness to pruning. The key observation is that, because of the wayInterleavers of size 2^(m) are constructed, for any window of size 2W, Wan integer, there is at most W indices that must be pruned in order toobtain an Interleaver of size N wherein 2^(m−1)<N≦2^(m). An Interleaverof 17 elements is obtained by pruning the interleaver of equation (18).

$\begin{matrix}{{7\mspace{14mu} 14\mspace{14mu} 3\mspace{14mu} 12\mspace{14mu} 6\mspace{14mu} 15\mspace{14mu} 1\mspace{14mu} 13\mspace{14mu} 5\mspace{14mu} 16\mspace{14mu} 9\mspace{14mu} 4\mspace{14mu} 11\mspace{14mu} 2\mspace{14mu} 8\mspace{14mu} 0\mspace{14mu} 10},} & (19)\end{matrix}$

resulting in an Interleaver matrix having elements of 7 14 3 12 6 15 113 5 16 9 4 11 2 8 0 10.

Thus, the previously discussed rules for good turbo interleaverdesign—Rules 1 and 2 given in equations (1) and (2)—can be generalizedto provide rules for the design of good turbo interleavers that arerobust to pruning. The modified rules are as follows:

Modified Rule 1: Minimize the occurrence of events.

|I[x]−I[x−j]|=7 where 7≦j≦14, j∈N   (20)

Modified Rule 2: If the first modified rule is satisfied with zerooccurrence, minimize the occurrence of events.

|I[x]−I[x−j]|=7, or   (21)

|I[x]−I[x−j]|=14 where 7≦j≦28, j∈N   (22)

A third Rule is also introduced: Rule 3: If the Modified Rule 1 andModified Rule 2 are satisfied with zero occurrence, maximize thevariable S such that, neighbor positions within a window size S are notmapped to neighbor positions within a window of size S.

Because of the key observation stated before the modified rules, it canbe seen that if an Interleaver of size 2^(M) satisfies Modified Rule 1and Modified Rule 2, then all of the Interleavers of size N(2^(m−1)<N≦2^(m)) obtained by pruning satisfy Rule 1 and Rule 2 statedat the beginning of this invention.

In designing preferred integer constants i_(o) and i_(b) for the GaloisField permutations to achieve pseudo-randomness in a Galois FieldInterleaver, performance of constructed Interleaver matrices aremeasured according to how well they meet the modified rules above.

Interleaver matrices of size 128, 256, 512, 1024, 2048 and 4096 areconstructed in accordance with the modified rules to yield near optimalInterleaver matrices of any size N where 64<N≦4096.

The primitive polynomials used to construct GF(c), where c is the numberof columns of the Interleaver matrix are as follows:

p(x)=x ⁴ +x+1 to construct GF(16)   (23)

p(x)=x ⁵ +x ²+1 to construct GF(32)

p(x)=x ⁶ +x+1 to construct GF(64)

p(x)=x ⁷ +x ³+1 to construct GF(128)

p(x)=x ⁸ +x ⁴ +x ³ +x ² +x to construct GF(256)

p(x)=x ⁹ +x ⁴+1 to construct GF(512)

Table 2 shows best values of i_(b) and i₀, as defined and determinedabove, for each row index i for each Interleaver matrix size specified.

TABLE 2 Galois Field (GF) Turbo Interleavers of Size 2^(m) = rxc RowInt. Size: Int. Size: Int. Size: Int. Size: Int. Size: Int. Size: Index128 = 8 × 16 256 = 8 × 32 512 = 16 × 32 1024 = 16 × 64 2048 = 32 × 644096 = 32 × 128 i i_(b) i₀ i_(b) i₀ i_(b) i₀ i_(b) i₀ i_(b) i₀ i_(b) i₀0 13 1 18 22 11 3 26 19 29 50 11 11 1 2 8 5 8 8 28 17 20 55 20 89 20 214 6 1 4 18 22 26 60 32 13 13 82 3 8 14 12 15 2 27 34 6 38 15 33 120 413 12 24 13 5 11 25 60 17 51 2 105 5 4 5 23 25 8 12 58 38 20 37 71 69 613 6 16 27 9 14 41 5 34 50 73 23 7 14 9 16 30 8 7 25 40 4 24 70 87 8 249 32 55 32 36 64 72 9 14 16 26 31 8 0 95 73 10 28 6 38 16 31 25 14 36 1111 17 50 28 61 37 108 102 12 9 2 23 46 62 38 21 64 13 3 24 22 40 25 2767 109 14 2 3 55 32 10 41 14 42 15 11 14 19 21 43 51 106 27 16 37 5 6364 17 10 43 17 13 16 41 54 65 5 19 26 4 62 46 20 10 44 116 111 21 40 1912 68 22 17 26 65 48 23 44 60 53 3 24 16 23 66 60 25 19 39 47 90 26 3858 126 59 27 47 54 115 1 28 13 38 113 38 29 46 7 12 9 30 46 22 57 75 3117 13 55 6

In all cases of Interleavers and associated matrices designed from theabove constants in accordance with the invention, Modified Rule 1 andModified Rule 2 is completely satisfied except for a small number of

In one embodiment, a computer search determines the constants such thatthe Modified Rules are satisfied.

Referring to FIG. 4, simulation results are shown for a random, S-randomand the new Galois Field Interleaver in accordance with the presentinvention of size 1024 over AWGN channel with overall Turbo code rate ½,wherein the encoder consists of eight-state constituent encoders withthe transfer function:

$\begin{matrix}{{G(D)} = \left\lbrack {1,\frac{1 + D + D^{3}}{1 + D^{2} + D^{3}}} \right\rbrack} & (24)\end{matrix}$

Curve 410 of FIG. 4 shows that with four (4) decoder iterations, theGalois Field (GF) Interleaver has about a 0.1 dB gain with respect to acomparable S-Random Interleaver (S=12) at a bit error rate of 10⁻⁵. Acomparable random Interleaver has the worst performance. For Frame ErrorRate performance, curve 420 shows that a (GF) Interleaver is also thebest performing Interleaver.

FIG. 5 illustrates the corresponding performance for eight (8) decoderiterations. Curve 510 illustrates the Bit Error Rate performancecompared to the others. Curve 520 illustrates the Frame Error Rateperformance compared to the others. Results are consistent with FIG. 4.

FIG. 6 illustrates the performance of Turbo Interleavers of size 1152over AWGN channel with overall Turbo code rate ⅓, and with four (4)decoder iterations. The GF Interleaver of size 1152 is formed from a(GF) Interleaver of size 2048 in accordance with this present invention.In this case, performance curves 310 for Bit Error Rate and 320 forFrame Error Rate illustrate that (GF) Interleavers have comparableperformance with S-random Interleaver (S=13).

The illustrative design of Table 2 can be simplified by furtherrestricting the choice of parameters. For example, the hardwareimplementation as well as storage requirements are reduced if theparameter i_(b) is made constant and equal to 1. In this case, theparameters describing the constituent permutations to be applied withineach row R should be re-optimized with respect to Modified Rules 1 and2.

Table 4 shows the near optimal values of integer constant i₀) for eachrow index i and each Interleaver matrix size, if i_(b) is 1.

TABLE 4 Simplified Galios Field (GF) Turbo Interleavers of Size 2^(m) =rxc Int. Size: Int. Size: Int. Size: Int. Size: Int. Size: Int. Size:Int. Size: Int. Size: Row 256 = 512 = 10 × 24 = 2048 4096 = 8192 = 16384= 32768 = Index 8 × 32 16 × 32 16 × 64 32 × 64 32 × 128 64 × 256 64 ×256 64 × 512 i i₀ i₀ i₀ i₀ i₀ i₀ i₀ i₀ 0 1 20 44 54 33 42 209 282 1 4 140 52 84 89 35 107 2 22 14 3 38 56 3 91 226 3 3 17 10 42 110 17 68 132 45 5 8 5 92 32 242 391 5 29 21 45 13 19 24 252 362 6 28 24 7 45 62 23 131119 7 9 28 28 60 50 11 208 139 8 19 37 25 45 109 29 129 9 12 2 2 73 104175 446 10 30 17 27 14 86 37 65 11 16 14 59 104 54 233 207 12 29 55 5378 60 12 95 13 2 48 29 103 52 141 153 14 25 19 33 98 55 196 208 15 23 1220 59 95 239 399 16 61 67 102 160 20 17 4 46 51 150 51 18 57 0 72 20 819 1 74 27 62 77 20 30 38 107 240 385 21 58 36 33 220 422 22 35 124 11042 434 23 40 61 29 235 509 24 7 48 28 21 168 25 3 112 94 58 273 26 6 111105 10 81 27 41 87 5 119 w465 28 18 49 63 115 219 29 28 125 16 61 319 3032 44 64 176 177 31 48 93 81 228 140 32 41 107 60 33 112 14 288 34 96 8768 35 100 125 80 36 124 24 183 37 4 254 293 38 7 179 121 39 45 127 13640 10 33 96 41 74 149 186 42 111 226 269 43 84 36 150 44 20 80 335 45 75109 138 46 26 11 41 47 117 133 144 48 93 210 202 49 103 117 218 50 0 30357 51 66 40 238 52 78 138 22 53 92 79 299 54 37 16 297 55 91 216 468 5671 198 24 57 40 143 161 58 43 248 328 59 38 69 237 60 61 87 104 14 62 50203

While the invention herein disclosed has been described by means ofspecific embodiments and applications thereof, numerous modificationsand variations could be made thereto by those skilled in the art withoutdeparting from the scope of the invention set forth in the claims.

For example, in the preferred embodiments, the constituent permutationsapplied within each row of the interleaver matrix were based on discretelogarithms in a

Galois field. It is clear that this is only a particular example of abroad class of permutations based on Galois field arithmetic, which maybe employed in accordance herewith. Optimally, another closely relatedchoice would be to take a non-primitive element βεGF(C) ofmultiplicative order ord(β) and define

π_(i)(J)=log(β^(io)+β^(j)), (j=0, 1, 2, . . . , ord(β)−1)

to produce a constituent permutation of length ord (β)−1. Alternatively,a logarithm of a different linear or affine function of β^(j) may beemployed. More generally, one could take permutation mapping data atposition i=0, 1, 2, . . . , ord(β)−1 to a new position

π_(i)(j)=f(β^(j)),

wherein f is any integer-valued function acting on finite field GF(C)and β is a non-zero element in GF(C) of multiplicative order ord(β).

In yet another alternative embodiment, the finite field(s) need not bebinary-that is C need not be a power of 2.

1. A turbo encoder comprising: a first constituent encoder configured toencode input bits in order to generate first parity bits; an internalinterleaver configured to interleave the input bits based on aninterleaver matrix in order to generate interleaved bits; and a secondconstituent encoder configured to encode the interleaved bits in orderto generate second parity bits, wherein a size of the interleaver matrixis determined based on a total number of the input bits, and whereingenerating the interleaved bits comprises pruning at least one bitoutput by the interleaver matrix after writing the input bits into theinterleaver matrix if a size of the interleaver matrix is larger thanthe total number of the input bits.
 2. The turbo encoder of claim 1,wherein the size of the interleaver matrix is obtained by multiplying anumber of rows of the interleaver matrix by a number of columns of theinterleaver matrix.
 3. The turbo encoder of claim 1, wherein the totalnumber of the input bits is equal to a total number of the interleavedbits.
 4. The turbo encoder of claim 1, wherein the internal interleaveris further configured to perform two-dimensional permutation afterwriting the input bits into the interleaver matrix.
 5. The turbo encoderof claim 1, wherein each of the first and second constituent encoders isan 8-state constituent encoder.
 6. The turbo encoder of claim 1, whereininitial values of shift registers in the first and second constituentencoders are set to all zeros.